Bochner–Martinelli formula

Remarks

The forms $d\bar\zeta_k \wedge d\zeta_k$ have degree $2$, so the order of the factors in the product does not matter. For fixed $z$, the kernel satisfies $d_\zeta \, \omega(\zeta, z) = 0$ on $\zeta \ne z$. Apply Stokes' theorem on $D$ minus a small ball around $z$ and let the radius tend to $0$; the small sphere contributes $f(z)$ in the limit. If $f$ is holomorphic near $\overline{D}$, then $\bar\partial f = 0$ and $f(z) = \int_{\partial D} f(\zeta) \, \omega(\zeta, z)$. For $n = 1$, $\omega(\zeta, z) = \frac{1}{2\pi i} \frac{d\zeta}{\zeta - z}$ and the formula is the Cauchy–Pompeiu formula, which contains the Cauchy integral formula. For $n \ge 2$ the kernel is not holomorphic in $z$, unlike the Cauchy kernel of the polydisc, but it works on every bounded domain with $C^1$ boundary.

Going deeper

For a student

The kernel is built from the fundamental solution of the Laplacian on $\R^{2n}$, which is why it is not holomorphic in $z$ when $n \ge 2$. The formula is a main tool for solving the equation $\bar\partial u = g$ and for proving extension results such as the Hartogs–Bochner theorem. It is the special case $\eta_j = (\bar\zeta_j - \bar z_j)/\abs{\zeta - z}^2$ of the Cauchy–Fantappiè formula for holomorphic functions.